The existence of globally smooth solutions of three dimensional Nernst-Planck equations with arbitrary large data coupled to Stokes equations driven by the Lorentz force has been obtained only recently in [1]}, [2]}.
Replacing viscous dissipation by friction diminishes the regularizing effect by two differential orders. In this paper we use the nonlinear structure of the NP equations coupled to Darcy's law, in order to prove the global regularity. The construction of solutions can be achieved by a two-tier approximation procedure as in [3]}, [4]}. The two-tier approximation scheme is needed because the global a priori \(L^p\) -estimate (REF )(i) is a consequence of the special nonlinear structure of (REF )–(), and standard approximation schemes do not preserve the resulting nonlinear dissipation (see (REF )). In this paper we focus on the a priori estimates, which are the heart of the matter.
